Large Sample Sieve Estimation of Semi-Nonparametric Models

Often researchers find parametric models restrictive and sensitive to deviations from the parametric specifications; semi-nonparametric models are more flexible and robust, but lead to other complications such as introducing infinite-dimensional parameter spaces that may not be compact and the optimization problem may no longer be well-posed. The method of sieves provides one way to tackle such difficulties by optimizing an empirical criterion over a sequence of approximating parameter spaces (i.e., sieves); the sieves are less complex but are dense in the original space and the resulting optimization problem becomes well-posed. With different choices of criteria and sieves, the method of sieves is very flexible in estimating complicated semi-nonparametric models with (or without) endogeneity and latent heterogeneity. It can easily incorporate prior information and constraints, often derived from economic theory, such as monotonicity, convexity, additivity, multiplicity, exclusion and nonnegativity. It can simultaneously estimate the parametric and nonparametric parts in semi-nonparametric models, typically with optimal convergence rates for both parts. This chapter describes estimation of semi-nonparametric econometric models via the method of sieves. We present some general results on the large sample properties of the sieve estimates, including consistency of the sieve extremum estimates, convergence rates of the sieve M-estimates, pointwise normality of series estimates of regression functions, root-n asymptotic normality and efficiency of sieve estimates of smooth functionals of infinite-dimensional parameters. Examples are used to illustrate the general results.